Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed 1.066459. We then consider the sequence formed by the minimal kth eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as k goes to infinity.